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Dual numbers were introduced in 1873 by William Clifford, and were used at the beginning of the twentieth century by the German mathematician Eduard Study, who used them to represent the dual angle which measures the relative position of two skew lines in space. Study defined a dual angle as ϑ+dε{\displaystyle \vartheta +d\varepsilon }, where ϑ{\displaystyle \vartheta } is the angle between the directions of two lines in three-dimensional space and d{\displaystyle d} is a distance between them. The n-dimensional generalization, the Grassmann number, was introduced by Hermann Grassmann in the late 19th century.

so the exponential map applied to the ε-axis covers only half the "circle".

Let z = a + b ε. If a ≠ 0 and m = b /a, then z = a(1 + m ε) is the polar decomposition of the dual number z, and the slopem is its angular part.
The concept of a rotation in the dual number plane is equivalent to a vertical shear mapping since (1 + p ε)(1 + q ε) = 1 + (p+q) ε.

relates the resting coordinates system to a moving frame of reference of velocityv.
With dual numbers t + x ε representing events along one space dimension and time,
the same transformation is effected with multiplication by (1 + v ε).

Given two dual numbers p, and q, they determine the set of z such that the difference in slopes ("Galilean angle") between the lines from z to p and q is constant. This set is a cycle in the dual number plane; since the equation setting the difference in slopes of the lines to a constant is a quadratic equation in the real part of z, a cycle is a parabola. The "cyclic rotation" of the dual number plane occurs as a motion of the projective line over dual numbers. According to Yaglom (pp. 92,3), the cycle Z = {z : y = α x2} is invariant under the composition of the shear

The image of X in the quotient is the unit ε. With this description, it is clear that the dual numbers form a commutative ring with characteristic 0. The inherited multiplication gives the dual numbers the structure of a commutative and associative algebra over the reals of dimension two. The algebra is not a division algebra or field since the elements of the form 0 + bε are not invertible. All elements of this form are zero divisors (also see the section "Division"). The algebra of dual numbers is isomorphic to the exterior algebra of R1{\displaystyle \mathbb {R} ^{1}}.

This construction can be carried out more generally: for a commutative ringR one can define the dual numbers over R as the quotient of the polynomial ringR[X] by the ideal (X2): the image of X then has square equal to zero and corresponds to the element ε from above.

Over any ring R, the dual number a + bε is a unit (i.e. multiplicatively invertible) if and only if a is a unit in R. In this case, the inverse of a + bε is a−1 − ba−2ε. As a consequence, we see that the dual numbers over any field (or any commutative local ring) form a local ring, its maximal ideal being the principal ideal generated by ε.

A narrower generalization is that of introducing n anti-commuting generators; these are the Grassmann numbers or supernumbers, discussed below.

Dual numbers find applications in physics, where they constitute one of the simplest non-trivial examples of a superspace. Equivalently, they are supernumbers with just one generator; supernumbers generalize the concept to n distinct generators ε, each anti-commuting, possibly taking n to infinity. Superspace generalizes supernumbers slightly, by allowing multiple commuting dimensions.

The motivation for introducing dual numbers into physics follows from the Pauli exclusion principle for fermions. The direction along ε is termed the "fermionic" direction, and the real component is termed the "bosonic" direction. The fermionic direction earns this name from the fact that fermions obey the Pauli exclusion principle: under the exchange of coordinates, the quantum mechanical wave function changes sign, and thus vanishes if two coordinates are brought together; this physical idea is captured by the algebraic relation ε2 = 0.

One application of dual numbers is automatic differentiation. Consider the real dual numbers above. Given any real polynomial P(x) = p0+p1x+p2x2+...+pnxn, it is straightforward to extend the domain of this polynomial from the reals to the dual numbers. Then we have this result:

where P′{\displaystyle P^{\prime }} is the derivative of P{\displaystyle P}.[2]

By computing over the dual numbers, rather than over the reals, we can use this to compute derivatives of polynomials.

More generally, we can extend any (analytic) real function to the dual numbers by looking at its Taylor series: f(a+bε)=∑n=0∞f(n)(a)bnεnn!=f(a)+bf′(a)ε{\displaystyle f(a+b\varepsilon )=\sum _{n=0}^{\infty }{{f^{(n)}(a)b^{n}\varepsilon ^{n}} \over {n!}}=f(a)+bf'(a)\varepsilon }, since any terms of involving ε2{\displaystyle \varepsilon ^{2}} or greater are trivially 0{\displaystyle 0} by the definition of ε{\displaystyle \varepsilon }.
By computing compositions of these functions over the dual numbers and examining the coefficient of ε in the result we find we have automatically computed the derivative of the composition.

A similar method works for polynomials of n variables, using the exterior algebra of an n-dimensional vector space.

Division of dual numbers is defined when the real part of the denominator is non-zero. The division process is analogous to complex division in that the denominator is multiplied by its conjugate in order to cancel the non-real parts.

This means that the non-real part of the "quotient" is arbitrary and division is therefore not defined for purely nonreal dual numbers. Indeed, they are (trivially) zero divisors and clearly form an ideal of the associative algebra (and thus ring) of the dual numbers.

Suppose D is the ring of dual numbers x + y ε and U is the subset with x ≠ 0. Then U is the group of units of D. Let B = {(a,b) in D x D : a ∈ U or b ∈ U}. A relation is defined on B as follows: (a,b) ~ (c,d) when there is a u in U such that ua=c and ub=d. This relation is in fact an equivalence relation. The points of the projective line over D are equivalence classes in B under this relation: P(D) = B/ ~.

Consider the embedding D → P(D) by z → U(z,1) where U(z,1) is the equivalence class of (z,1). Then points U(1,n), n2 = 0, are in P(D) but are not the image of any point under the embedding. P(D) is projected onto a cylinder by projection: Take a cylinder tangent to the double number plane on the line {y ε: y ∈ ℝ}, ε2 = 0. Now take the opposite line on the cylinder for the axis of a pencil of planes. The planes intersecting the dual number plane and cylinder provide a correspondence of points between these surfaces. The plane parallel to the dual number plane corresponds to points U(1,n), n2 = 0 in the projective line over dual numbers.